Tour du Mont Blanc

I have done it - twice

2015 - Clockwise - The general advice on the Internet was that was the better direction to do it on a mountain bike. Such a load of crap. You push & hike up on the amazing singletrack to Col de la Croix du Bonhomme, and then do a mostly gravel road descent to Les Contamines.
On the other side of the mountain at Col de Ferret on the Swiss side you again climb some amazing singletrack, but then on the descent on the Italian side towards the refugio there are dozens of little stone water barriers (5-10cm) that are an absolute bitch to get around/hop over.
I did it with a friend, camping with hammocks and a tarp. Wild camping is allowed in France in the lower parts of the mountains. Bivouacing is generally fine anywhere if you move in the morning. Paid for a camping in Chamonix to use the shower.
Took us 6 days with a day and a half spent in the Les Houches & Les Tours bike parks.
My full gear (this means anything, besides me) was 27.183kg, of them the bike was 11.8Kg (160mm carbon hardtail). This includes a full face, shin and arm guards (we spent some time in the bikeparks).
Budget of €300 total for the two of us, sans travel costs to get to Courmayor.

2016 - Counterclockwise - this in my opinion is the much, much better direction. Climbs are not too bad and descents are superb.
This time we stayed at huts with the idea to carry much less weight, but didn't save that much actually - 12.00kg bike (beefier tires) + 12.228kg gear. Budget skyrocketed to €900 each, but this includes a good 4 or 5 days of bike parks @ Verbier & Portes du Soleil + train to and from the Portes (got to Verbier on the bikes via Orsieres.

We followed the classic TMB route both times. I have GPX tracks, if you want.